On sides $AB$ and $BC$ of an acute triangle $ABC$ two congruent rectangles $ABMN$ and $LBCK$ are constructed (outside of the triangle), so that $AB = LB$. Prove that straight lines $AL, CM$ and $NK$ intersect at the same point. [i](5 points)[/i]

Let $z_1, z_2, z_3$ be pairwise distinct complex numbers satisfying $|z_1| = |z_2| = |z_3| = 1$ and \[\frac{1}{2 + |z_1 + z_2|}+\frac{1}{2 + |z_2 + z_3|}+\frac{1}{2 + |z_3 + z_1|} =1.\] If the points $A(z_1),B(z_2),C(z_3)$ are vertices of an acute-angled triangle, prove that this triangle is equilateral.

In the coordinate plane, consider the set of all segments of integer lengths whose endpoints have integer coordinates. Prove that no two of these segments form an angle of $45^o$. Are there such segments in coordinate space?

Let $m$ and $n$ be distinct positive integers. Prove that $2^{2^m} + 1$ and $2^{2^n} + 1$ have no common divisor greater than $1$.

$f(x)$ is a decreasing function defined on $(0,+\infty)$, if $f(2a^2+a+1)<f(3a^2-4a+1)$, then the range value of $a$ is________.

Bassanio has three red coins, four yellow coins, and five blue coins. At any point, he may give Shylock any two coins of different colors in exchange for one coin of the other color; for example, he may give Shylock one red coin and one blue coin, and receive one yellow coin in return. Bassanio wishes to end with coins that are all the same color, and he wishes to do this while having as many coins as possible. How many coins will he end up with, and what color will they be?

Let $(x_1,y_1) = (0.8, 0.6)$ and let $x_{n+1} = x_n \cos y_n - y_n \sin y_n$ and $y_{n+1}= x_n \sin y_n + y_n \cos y_n$ for $n=1,2,3,\dots$. For each of $\lim_{n\to \infty} x_n$ and $\lim_{n \to \infty} y_n$, prove that the limit exists and find it or prove that the limit does not exist.

Let $ABCD$ be a cyclic quadrilateral with $AB=4,BC=5,CD=6,$ and $DA=7$. Let $A_1$ and $C_1$ be the feet of the perpendiculars from $A$ and $C$, respectively, to line $BD,$ and let $B_1$ and $D_1$ be the feet of the perpendiculars from $B$ and $D,$ respectively, to line $AC$. The perimeter of $A_1B_1C_1D_1$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Baron Munchausen took several cards and wrote a positive integer on each one (some numbers may be the same). The baron reports that he has used only two distinct digits to do that. He also reports that among the leftmost digits of the sums of integers on each pair of these cards there are all the digits from 1 to 9 . Can it occur that the baron is right? Maxim Didin

The ship [i]Meridiano do Bacalhau[/i] does its fishing business during $64$ days. Each day the capitain chooses a direction which may be either north or south and the ship sails that direction in that day. On the first day of business the ship sails $1$ mile, on the second day sails $2$ miles; generally, on the $n$-th day it sails $n$ miles. After of the $64$-th day, the ship was $2014$ miles north from its initial position. What is the greatest number of days that the ship could have sailed south?

An equilateral triangle $T$ with side $111$ is partitioned into small equilateral triangles with side $1$ using lines parallel to the sides of $T$. Every obtained point except the center of $T$ is marked. A set of marked points is called $\textit{linear}$ if the points lie on a line, parallel to a side of $T$ (among the drawn ones). In how many ways we can split the marked point into $111$ $\textit{linear}$ sets?

Define $S_r(n)$: digit sum of $n$ in base $r$. For example, $38=(1102)_3,S_3(38)=1+1+0+2=4$. Prove: [b](a)[/b] For any $r>2$, there exists prime $p$, for any positive intenger $n$, $S_{r}(n)\equiv n\mod p$. [b](b)[/b] For any $r>1$ and prime $p$, there exists infinitely many $n$, $S_{r}(n)\equiv n\mod p$.

Each point of the plane is coloured in one of two colours. Given an odd integer number $n \geq 3$, prove that there exist (at least) two similar triangles whose similitude ratio is $n$, each of which has a monochromatic vertex-set. [i]Vasile Pop[/i]

Find the real root of $x^5+5x^3+5x-1$. Hint: Let $x = u+k/u$.

Prove that every tetrahedron has a vertex whose three edges have the right lengths to form a triangle.

Dirock has a very neat rectangular backyard that can be represented as a $32\times 32$ grid of unit squares. The rows and columns are each numbered $1,2,\ldots, 32$. Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by $3$. Dirock would like to build a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of the yard such that [list] [*] The fence does not pass through any grid squares containing rocks; [*] The interior of the fence contains exactly 5 rocks. [/list] In how many ways can this be done? [i]Ray Li[/i]

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$: $$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$ [i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]

Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides. Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles. [i]UK - Sahl Khan[/i]

Hexagon $ABCDEF$ is inscribed in a circle $O$. Let $BD \cap CF = G, AC \cap BE = H, AD \cap CE = I$ Following conditions are satisfied. $BD \perp CF , CI=AI$ Prove that $CH=AH+DE$ is equivalent to $GH \times BD = BC \times DE$

Let $\triangle ABC$ be an acute-angled triangle with $AB>AC$, $O$ be its circumcenter and $D$ the midpoint of side $BC$. The circle with diameter $AD$ meets sides $AB,AC$ again at points $E,F$ respectively. The line passing through $D$ parallel to $AO$ meets $EF$ at $M$. Show that $EM=MF$.

The convex quadrilateral $ABCD$ is given, $AB<BC$ and $AD<CD$. $P,Q$ are points on $BC$ and $CD$ respectively such that $PB=AB$ and $QD=AD$. $M$ is midpoint of $PQ$. We assume that $\angle BMD=90^{\circ}$, prove that $ABCD$ is cyclic.

An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices $25$ times. Is it possible that it has visited each of the other $7$ vertices exactly $20$ times? (S Tokarev)

Find $ a,b,c \in N$ such that $ ab$ divides $ a^2\plus{}b^2\plus{}1$.

Given a finite sequence with terms in the set $A=\{0,1,…,121\}$ , it is allowed to replace each term by a number from the set $A$ so that like terms are replaced by like numbers, and different terms by different numbers. (Terms may remain without replacement.) The objective is to obtain, from a given sequence, through several such changes, a new sequence with sum divisible by $121$ . Show that it is possible to achieve the objective for every initial sequence. [hide=original wording]Dada una secuencia finita con términos en el conjunto A={0,1,…,121} , está permitido reemplazar cada término por un número del conjunto A de modo que términos iguales se reemplacen por números iguales, y términos distintos por números distintos. (Pueden quedar términos sin reemplazar.) El objetivo es obtener, a partir de una sucesión dada, mediante varios de tales cambios, una nueva sucesión con suma divisible por 121 . Demostrar que es posible lograr el objetivo para toda sucesión inicial.[/hide]

Triangle $ABC$ is equilateral with side length $12$. Point $D$ is the midpoint of side $\overline{BC}$. Circles $A$ and $D$ intersect at the midpoints of side $AB$ and $AC$. Point $E$ lies on segment $\overline{AD}$ and circle $E$ is tangent to circles $A$ and $D$. Compute the radius of circle $E$. [i]2015 CCA Math Bonanza Individual Round #5[/i]